Let $(a_n)$ be a sequence. Consider the sequence of even-indexed terms, $(a_{2n})$, and the sequence of odd-indexed terms, $(a_{2n+1})$. Prove that if both $(a_{2n})$ and $(a_{2n+1})$ converge to $L$, then $(a_n)$ converges to $L$.

Let $q_n=\displaystyle\frac{f_n}{f_{n+1}}$, where $f_n$ is the $n$-th Fibonacci number. Show that the sequence $(q_n)$ converges. What value does it converge to?